Consider the \(N\times N\)  matrix \(B_{N}=\left( b_{s,k}\right) _{\left(s,k\right) =\left( 0,0\right) }^{\left( N-1,N-1\right) }\) . Then (REF ) implies that \(\det B_{N}=1,\)  which means that this matrix is invertible.
In fact, the existence of the matrix \(B_{N}^{-1}\)  for each \(N\ge 1\)  is the
key property why the basis \(\lbrace Q_{n}\left( \alpha \right) \rbrace _{n=0}^{\infty }\) 
was originally constructed in [1]}. Indeed, consider, for example
either the basis of standard orthonormal polynomials or the basis of
trigonometric functions. In each of these, the first function is an
identical constant, which means that the first raw of an analog of the
matrix \(B_{N}\)  is zero.
